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Discrete Math and Analyzing Social Graphs
Week 3
Probabilities and combinatorics

QuizQuiz • 10 min10 minutes

Probabilities and combinatorics

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Probabilities and combinatorics

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Probabilities and combinatorics

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1.

Question 1

There is a box with 5 green and 3 yellow balls. Two balls where chosen randomly (without replacement). What is the probability that both balls are green?

1 / 1 point

$\frac{\binom{8}{2}}{\binom{5}{2}}$

$\frac{\binom{5}{2}}{\binom{8}{2}}$

$\frac{\binom{5}{2}}{\binom{5}{3}}$

$\frac{\binom{8}{3}}{\binom{8}{5}}$

Correct

Correct: the denominator is the number of all outcomes (select two balls out of eight) and the numerator is the number of all outcomes that satisfy our condition (select two balls out of five green).

2.

Question 2

There is a box with 5 green and 3 yellow balls. Two balls where chosen randomly (without replacement). What is the probability that one ball is green and another is yellow?

1 / 1 point

$\frac{\binom{8}{2}}{\binom{5}{1}\cdot \binom{3}{1}}$

$\frac{\binom{5}{1}\cdot\binom{3}{1}}{\binom{8}{2}}$

$\displaystyle \frac{\binom{5}{1} + \binom{3}{1}}{\binom{8}{2}}$

$\frac{\binom{8}{5}\cdot \binom{8}{3}}{\binom{8}{2}}$

Correct

Let's count outcomes that satisfy the conditon. We have to select one green ball out of five green balls, that's $\binom{5}{1}$ . We also have to select one yellow ball out of three yellow balls, that's $\binom{3}{1}$ . Then we use product rule to find many ways there exist to select one green ball and one yellow ball. That's numerator. In the denominator, we have the number of all ways to select two balls out of eight.